Fifteen years ago, Ana Patricia Botín was pushed out of her senior job at Banco Santander SA by the bank's chairman, who also happened to be her father. On Wednesday, Ms. Botín's Shakespearean career came full circle, with her being named chairman after her father's sudden death.

Now the question is whether the 53-year-old Ms. Botín, a hard-nosed, detail-obsessed executive who was groomed since childhood to take Santander's reins,...